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Home Search Collections Journals About Contact us My IOPscience Optical measurement of the gas number density in a Fabry–Perot cavity This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 Meas. Sci. Technol. 24 105207 (http://iopscience.iop.org/0957-0233/24/10/105207) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.239.110.148 The article was downloaded on 17/09/2013 at 11:57 Please note that terms and conditions apply. IOP PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY doi:10.1088/0957-0233/24/10/105207 Meas. Sci. Technol. 24 (2013) 105207 (5pp) Optical measurement of the gas number density in a Fabry–Perot cavity Isak Silander 1 , Martin Zelan 2 , Ove Axner 1 , Fredrik Arrhén 2 , Leslie Pendrill 2 and Aleksandra Foltynowicz 1 1 2 Department of Physics, Umeå University, SE-901 87 Umeå, Sweden SP Swedish Technical Research Institute, SE-50 115 Borås, Sweden E-mail: [email protected] and [email protected] Received 8 May 2013, in final form 12 August 2013 Published 17 September 2013 Online at stacks.iop.org/MST/24/105207 Abstract An optical method for measuring the gas density by monitoring the refractive index inside a high-finesse Fabry–Perot cavity is presented. The frequency of a narrow linewidth Er:fiber laser, locked to a mode of the cavity, is measured with the help of an optical frequency comb while the gas density inside the cavity changes. A resolution of 1.4 × 10−6 mol m−3 is achieved in 3 s for nitrogen, which allows measurement of a relative gas density change of 3.4 × 10−8 at atmospheric pressure. Keywords: laser refractometry, gas density measurement, metrology, optical frequency comb, Fabry–Perot cavity (Some figures may appear in colour only in the online journal) 1. Introduction index [4], which, in turn, has spurred significant advances in calculations of atomic properties such as refractivity and polarizability [5]. Optical refractometry, and in particular laser refractometry, has been successfully used to measure the absolute refractive index of dry gas to very high accuracy [6, 7]. It has also been applied to measurements of the density of air in order to compensate for the buoyancy in precision mass metrology [8, 9]. Interferometry with optical frequency combs (OFCs) has been employed for measurements of the refractive index of air and other gases [10–12]. We demonstrate a highly sensitive technique for the measurement of the number density of a gas inside a highfinesse Fabry–Perot cavity, based on an earlier feasibility study [13, 14]. In this method, a narrow linewidth fiber laser is locked to a mode of the cavity and the change in laser frequency caused by the shift of the cavity mode with changing refractive index is monitored with the use of an OFC. The technique provides fast time response where the achieved resolution in density determination is limited by the OFC reference and is independent of the absolute gas pressure. This makes the technique particularly appealing in higher density ranges, where the conventional techniques lack accuracy. Measurement of changes of gas density has several applications, where the most straightforward are for gas flow and leak detection and calibration. Conventional methods for the determination of gas density variation involve a measurement of changes of the pressure (under constant volume) or the volume (under constant pressure) [1]. In both these cases, the assessment relies on the use of the equation of state of a gas, whereby it is limited by the intrinsic accuracy of either the pressure gauge or accuracy of temperature measurement and control. As a consequence, the accuracy of conventional techniques is insufficient for applications aiming to detect small leaks or flows under atmospheric (or higher) pressure conditions [2]. By detecting instead the change of the refractive index, e.g., by monitoring the refractivity or the permittivity of the gas, an assessment of the gas density can be made independent (to first order) of both the pressure and the temperature. This enables single-parameter monitoring of gas density and its variation with significantly increased accuracy [3]. As a consequence, the last decade has seen the introduction of a number of new and improved optical and dielectric measurement techniques for measurements of the refractive 0957-0233/13/105207+05$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA Meas. Sci. Technol. 24 (2013) 105207 I Silander et al Figure 1. Experimental setup. A narrow linewidth fiber laser is locked to a mode of a Fabry–Perot cavity by the PDH technique. An electro-optic modulator (EOM) is phase-modulating the laser light, and an error signal is derived from the light reflected from the cavity monitored by a fast photodiode. The laser servo keeps the fiber laser locked to a cavity mode. The cavity is connected to a vacuum system in which the gas density can be controlled and the pressure can be measured. To measure the frequency of the fiber laser, the output from the laser is split and a beat note is created between the laser and an OFC. 2. Theory can be used for the determination of absolute gas density, with the accuracy to which the refractive index, molecular polarizability and optical frequency are known. The main experimental uncertainty originates from the measurement of the optical frequencies, and it has two contributions. The first comes from the finite resolution of the frequency measurement, δν, determined by the linewidth of the continuous wave (cw) laser or the frequency reference (whichever is larger). The second comes from the shift of the cavity mode frequencies due to the thermal drift of the cavity length, ∂L/∂T = αL, where α is the thermal expansion coefficient of the cavity spacer. Including the temperature and pressure dependence of molar polarizability that originates from the deviation from the ideal gas law, the accuracy of the determination of the gas density is given by 2 2 2δν 2 ν ∂A ν ∂A 2n δT + δp . + αδT + δρ ≈ 3A ν0 Aν0 ∂T Aν0 ∂ p (6) The gas number density, ρ, defined as N/V , where N is the amount of gas (in mol) and V is the volume (in m3), is related to the index of refraction (the optical refractivity), n, as 2 (n − 1), (1) ρ= 3A where A is the molar polarizability (in m3 mol–1) [8, 15]. The refractive index can be assessed accurately by placing the gas sample inside a Fabry–Perot resonator and measuring the frequency of a given mode of the cavity. The frequency of the qth cavity mode can be expressed as cq , (2) ν= 2Ln where c is the speed of light in vacuum, L is the cavity length and q is the mode number (under the assumption that the reflection phase shift due to mirror coating dispersion can be neglected, which is valid when operating near the design wavelength of the cavity mirrors). Solving equation (2) for n and differentiating with respect to ν shows that a change in the refractive index can be related to a change in the frequency of the cavity mode as n dn =− . (3) dν ν This implies that the change of gas density can be assessed by the measurement of the change in optical frequency of a cavity mode according to dρ dn 2 n dρ = =− . (4) dν dn dν 3A ν Integrating both sides of the equation yields 2n 2n ν 2n , (5) ρ − ρ0 = − ln(ν) + ln(ν0 ) ≈ − 3A 3A 3A ν0 where ν0 is the optical frequency corresponding to the reference density ρ0 , and the last step is valid for ν = ν − ν0 < ν0 . By setting the reference density to zero (corresponding to an evacuated cavity), the above equation 3. Experimental setup The experimental setup is illustrated in figure 1. A narrow linewidth Er:fiber laser (Koheras, Adjustik E15) operating around 1531 nm is locked using the Pound–Drever–Hall (PDH) [16] technique to a Fabry–Perot cavity with a finesse of 7400 and a free spectral range of 428 MHz. The cavity mirrors are glued to a 35 cm spacer made of Zerodur placed directly on an optical table, which, in turn, is placed in a temperaturestabilized room. The temperature of the room, optical table and the Zerodur spacer is monitored by PT-100 sensors (Pico Technology, PT-104). The cavity is connected to a vacuum system, which allows changes of the amount of gas in the chamber as well as continuous measurement of the pressure inside the cavity by the use of a pressure gauge (Oerlikon Leybold Vacuum, Ceravac CTR91-1Torr) with 1 s resolution. 2 I Silander et al 35 54 28 58 21 62 Frequency [MHz] Pressure [Pa] The sidebands for the PDH locking are created at 20 MHz with the use of a fiber-coupled LiNbO3 electrooptic modulator (EOM, General Photonics, LPM-001-15) and a fixed frequency source. The PDH error signal is obtained by phase sensitive detection of light intensity in cavity reflection using a high-bandwidth photo-detector. The bandwidth of the PDH servo loop is 10 kHz, limited by the resonance of the PZT in the fiber stretcher (at 35 kHz) to which the correction feedback is applied [17]. Since the linewidth of the fiber laser is narrow (less than 1 kHz @ 1 ms), such a bandwidth is sufficient to keep the laser locked to the center of the cavity mode, whose linewidth is 29 kHz (HWHM). The optical frequency of the cw laser is measured using an Er:fiber OFC (MenloSystems, FC-1500-250-WG) as a reference. A fraction of the cw light is split off with a fiber-splitter and used to create a beat note with the output of the frequency comb. The repetition rate and the carrier envelope offset frequency of the comb are stabilized to a GPS referenced rubidium oscillator (Symmetricom, TSC4410A), and the values of the two parameters are chosen so that the frequency of the beat note between the cw laser and the nearest comb line is around 60 MHz. The number of the closest comb line is determined by measuring the frequency of the cw laser with a wavemeter (Burleigh, WA-1500), which provides 30 MHz resolution. The sign of the beat note is determined by changing the repetition rate and carrier envelope offset frequency in a controlled way and observing the sign of the change of the beat note. The frequency of the optical beat note is monitored using either a counter integrated in the frequency comb system with 1 Hz sampling rate or with a fast external counter (1 kHz sampling rate, Agilent 53230A). 14 7 70 66 65 60 70 55 0 0 0 0.8 Frequency [MHz] Meas. Sci. Technol. 24 (2013) 105207 5 10 15 20 Pressure [Pa] 1.6 Time [min] 25 30 74 3.2 2.4 Figure 2. Time dependence of pressure (solid red lower curve, left axis) and optical frequency (dashed blue upper curve, right axis) as the number density of nitrogen in the cavity is varied. The curves are offset for clarity. The inset shows the optical beat note frequency as a function of pressure with a linear fit. 4 Allan deviation [Hz] 10 4. Results 3 10 To test the performance of the method, we repeatedly filled the cavity with pure nitrogen to a given pressure and quickly evacuated it, while simultaneously monitoring the intracavity pressure and beat frequency between the cw laser and the OFC. Figure 2 shows the result for one cycle of filling and evacuating the cavity; the solid red curve displays the pressure in the chamber, the dashed blue curve—the beat note frequency. As expected, the two curves, offset for clarity, have the same general form. A closer scrutiny of the data showed that when the cavity was repeatedly filled and emptied, the laser frequency corresponding to the evacuated chamber was drifting slowly. This was caused by thermal expansion of the cavity spacer and hysteresis in its compressibility [6]. To correct for this drift in a series of repeated measurements, we fitted a third-order polynomial to the vacuum frequencies, ν0 , in between consecutive fillings and subtracted it from the measured data. The resulting dependence between the corrected frequency and pressure is shown by the inset in figure 2, where the solid line represents a linear fit with a slope of −519.26(7) kHz Pa–1. This clearly illustrates that the measured beat frequency has a linear dependence on pressure, within the accuracy of the pressure gauge. To investigate the accuracy as well as the long-term stability of the system, we measured the optical beat note 10 -1 10 0 1 10 Integration time [s] 2 10 3 10 Figure 3. The Allan deviation of the optical beat note between the laser locked to the evacuated cavity and the OFC. frequency for 2 h while keeping the cavity at vacuum. Under these conditions, the experimental uncertainty in density, equation (6), can be rewritten in terms of the measured frequency as 2δν 2 δνbeatnote ≈ ν0 + (αδT )2 . (7) ν0 Figure 3 shows the Allan deviation of the beat note measurement. At integration times up to 3 s, the data follow white noise behavior. At 1 s, the uncertainty is 1.4 kHz, which corresponds to a relative frequency uncertainty, 2δν/ν0 , of 7 × 10−12. This implies that at short integration times, the uncertainty is limited by the stability of the Rb oscillator to which the frequency comb is stabilized (specified to be 5 × 10−12 at 1 s). The drift at longer integration times is caused by thermal expansion of the cavity spacer. The temperature of the cavity spacer drifted at a rate of 0.01 K h−1 during the measurement. This change in spacer temperature results in 3 Meas. Sci. Technol. 24 (2013) 105207 I Silander et al a frequency drift of 3.2 kHz in 100 s (assuming a thermal expansion coefficient of Zerodur, α, of 6 × 10−8 K−1), in agreement with the measured Allan deviation. The highest resolution in the determination of the optical beat note frequency is obtained at an integration time of 3 s and is equal to 900 Hz. Provided that the temperature and pressure dependence of molar polarizability can be neglected, the frequency uncertainty can be recalculated to the minimum change of gas density using the first term in equation (6). The value of the molar polarizability of nitrogen at atmospheric pressure and 23 ◦ C is calculated from the definition, A = 2 (n − 1) RT Z/3p (where R is the molar gas constant, and Z is the compressibility), and equations (1), (2) and (4) in [18] to be 4.40 × 10−6 m3 mol−1. Thus the minimum change of gas density that can be resolved with the optical technique is 1.4 × 10−6 mol m−3 in 3 s, which corresponds to a relative accuracy in the density determination of 3.4 × 10−8 at atmospheric pressure. This result is valid provided that the error contributions from the temperature and pressure dependence of molar polarizability, i.e. the third and fourth terms in equation (6), are lower than the first term. The two terms, (∂A/∂T )/A and (∂A/∂ p)/A, are calculated by differentiating the definition of A to be −1 × 10−5 K−1 (at atmospheric pressure) and 2.6 × 10−9 Pa−1, respectively. Thus the variation of molar polarizability is negligible compared to the frequency uncertainty at 3 s if the temperature is constant to within 3 mK and the pressure to within 13 Pa. It is noteworthy that the accuracy of the optical technique is significantly less affected by changes in temperature and pressure than the conventional technique of determining gas density based on the equation of state of the gas. The accuracy of the conventional technique is directly dependent on the accuracy to which the temperature and pressure can be controlled and measured. Since the resolution of a pressure gauge is usually a fraction of its measurement range, typically of the order of 10−4, the gas number density assessment cannot be made with relative accuracy better than this fraction. Thus the presented optical technique has the potential to supersede the conventional technique by orders of magnitude. lasers locked to the two cavities [8, 9]. In an even simpler approach, one laser can be locked to both cavities with the aid of an acousto-optical modulator. The developed method can find application in several areas where the gas density change is the main observable. The most straightforward are the measurement of gas flows and the calibration of gas flow meters by monitoring the increase (or decrease) of gas density to/from a known volume. With the possibility of using differently sized volumes, the method can be tuned for maximal accuracy, with large volumes used for large flows and small volumes for small flows. By measuring the filling of an initially empty volume, the method can also be used in applications where the absolute gas density is of importance. Among them are currently employed methods for calibration of reference gas leaks for vacuum applications [19, 20], where one of the main uncertainties comes from the initial absolute pressure measurement. The method also gives a possibility of detecting small density changes even at high pressures, which might enable the detection of small critical leaks of unwanted (i.e. hazardous) gas at higher pressure. Besides applications in gas flow measurement and calibration, assessment of the gas number density can also be a promising start of a future redefinition of the current artifact-based pressure standard. A future standard based on the equation of state of a gas, often reduced to the ideal gas law, would allow a standard that is based on fundamental physical constants. Such a method will also make it possible to eliminate the use of hazardous mercury in the current worldwide spread methods for pressure standards. Acknowledgments This project was supported by the Faculty of Science and Technology, Umeå University, and the Swedish Research Council under the projects 621-2011-5123 and 621-20123650. References [1] Jackson C N Jr, Sherlock C N and Moore P O 1998 Leak Testing (Nondestructive Testing Handbook vol 1) (Columbus, OH: American Society of Nondestructive Testing) [2] Morgado I, Legras J C and Clodic D 2010 Primary standard for the calibration of refrigerant leak flow rates Metrologia 47 135–45 [3] Pendrill L R 2004 Refractometry and gas density Metrologia 41 S40–S51 [4] Schmidt J W, Gavioso R M, May E F and Moldover M R 2007 Polarizability of helium and gas metrology Phys. Rev. Lett. 98 254504 [5] Cencek W, Szalewicz K and Jeziorski B 2001 Breit–Pauli and direct perturbation theory calculations of relativistic helium polarizability Phys. Rev. Lett. 86 5675–8 [6] Egan P and Stone J A 2011 Absolute refractometry of dry gas to ± 3 parts in 109 Appl. 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By comparing the frequencies of two lasers locked to the modes of the two cavities, the common mode temperature drift can be subtracted. Such a method also eliminates the need for a stable absolute frequency source (OFC), since the change of the refractive index can be derived from the relative change of the frequencies of two 4 Meas. Sci. Technol. 24 (2013) 105207 I Silander et al [9] Fang H, Picard A and Juncar P 2002 A heterodyne refractometer for air index of refraction and air density measurements Rev. Sci. Instrum. 73 1934–8 [10] Zhang J, Lu Z H, Menegozzi B and Wang L J 2006 Application of frequency combs in the measurement of the refractive index of air Rev. Sci. Instrum. 77 083104 [11] Zhang J, Lu Z H and Wang L J 2008 Precision refractive index measurements of air, N2, O2, Ar, and CO2 with a frequency comb Appl. 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